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Lesson 20

UNIT FRACTIONS


In this Lesson, we will answer the following:

  1. What is a unit fraction?
  2. How can we express a whole number as a fraction?
  3. What do we mean by the complement of a proper fraction?

    Section 2

  4. What kind of fraction does "out of" indicate?

 1.   What is a unit fraction?
 
  A fraction whose numerator is 1.

 

A unit, recall, is whatever we call one. (Lesson 1.)  Each unit fraction is a part of number 1.

1
2
  is one half of 1.

1
3
  is the third part of 1.

1
4
  is the fourth part of 1.

And so on.


  Example 1.    In the fraction  4
5
, what number is the unit, and how many

of them are there?

Answer.  The denominator of a fraction names the unit -- the part of 1.  The numerator tells their number -- how many.

 In the fraction  4
5
, the unit is  1
5
.  And there are 4 of them.

 

  Example 2.   Let  1
3
 be the unit, and count to 2 1
3
.

We see that every fraction is a multiple of some unit fraction:

2
3
 =  2 ×  1
3
 =  1
3
 +  1
3
.
3
5
 =  3 ×  1
5
 =   1
5
 +  1
5
 +  1
5
.
  Example 3.   Add   2
8
 +  3
8
.
  Answer.   5
8
.
2 eighths + 3 eighths are 5 eighths. The unit is  1
8
.

This illustrates the following principle:

In addition and subtraction, the units must be the same.

We will see this in Lesson 24.  In any fraction, the denominator names the unit.

7
9
 −  3
9
  =   4
9

Example 4.   1 is how many fifths?

  Answer.    5
5
.
 
   
1
5
 is contained in 1 five times.

Similarly,

1 =  3
3
 =  4
4
 =  10
10

And so on.  We may write 1 with any denominator.  Which is to say, we may decompose 1 into any parts:  Halves, thirds, fourths, fifths, millionths.

  Example 5.   Add, and express the sum as an improper fraction:   5
9
 + 1.
  Answer.    5
9
 + 1 =  5
9
 +  9
9
 =  14
 9
.



 2.   How can we express a whole number as fraction?
  Multiply the whole number by the denominator. That product will be the numerator.

  Example 8.    2 =  2 × 5
   5
 =  10
 5
.
Since 1 =  5
5
, then 2 is twice as many fifths:  2 =  10
 5
.
  Example 9.    6 =  ?
3
  Answer.    6  =   6 × 3
   3
  =   18
 3
.
  Example 10.     How many times is  1
8
 contained in 5?  That is, 5 =  ?
8
.
  Answer.   5  =   40
 8
.

The complement of a proper fraction


 3.   What do we mean by the complement of a proper fraction?
  It is the proper fraction we must add in order to get 1.

  Example 11.    5
8
 + ? = 1
  Answer.   Since 1 =  8
8
, then  5
8
 +  3
8
 =  1.

Equivalently, since finding what number to add  is subtraction,

1 −  5
8
  =   3
8
.
3
8
 is called the complement of   5
8
.     3
8
 completes  5
8
 to make 1.
  Example 12.    How much is   1 −  1
3
?
  Answer.   1 −  1
3
  =   2
3
.
1
3
  plus   2
3
  =   3
3
, which is 1.
  Example 13.    1 −  2
5
 =  3
5
.
When we add   3
5
 to  2
5
,  we get 1.

  Example 14.   How much is 6 3
4
 −  1
4
 ?
  Answer.  6 2
4
.  The 6 is not affected.
  Example 15.   How much is 6 4
4
 −  1
4
 ?
  Answer.  6 3
4
.
But 6 4
4
 is 7.  That is,
7  −  1
4
  =  6 3
4

Look at the fact:

We are subtracting  1
4
 -- which is less than 1 -- from 7.  The answer
  therefore falls beween 6 and 7.  And  3
4
 is the complement of  1
4
.

Compare

1 −  1
4
:

In other words:

Whenever we subtract a proper fraction from a whole number greater than 1, the answer will be a mixed number which is one whole number less, and whose fraction is the complement of the proper fraction.

  Example 16. 5 −  1
3
 = 4 2
3

4 is one less than 5.  And  2
3
 is the complement of  1
3
.

On the other hand, we could say that we can only subtract thirds from thirds. Therefore we must create thirds by breaking off 1 from 5

  and calling it  3
3
.
5 −  1
3
 = 4 3
3
 −  1
3
 = 4 2
3
.
  Example 17.   9 −  2
5
  =  8 3
5
.

We could check this by adding:

8 3
5
 +  2
5
 = 9.

At this point, please "turn" the page and do some Problems.

or

Continue on to the next Section.


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